complex geometry - question about kahler cone of a compact kahler manifold

Hi to all!

I'm studying complex geometry from Huybrechts book "Complex Geometry"
and i have problems with an exercise, please can anyone help me?

I define the kahler cone of a compact kahler manifold X as the set

$K_X subseteq H^{(1,1)}(X)cap H^2(X,mathbb{R})$
of kahler classes. I have to prove that $K_X$ doesn't contain any line
of the form $alpha + t beta$ with $alpha , betain H^{(1,1)}(X)cap H^2(X,mathbb{R})$
and $betaneq 0$ (i identify classes with representatives).

This is what i thought: i know that a form $omega in H^{(1,1)}(X)cap H^2(X,mathbb{R})$
that is positive definite (locally of the form $frac{i}{2}sum_{i,j} h_{ij}(x)dz^iwedge d overline{z}^{j}$ and $(h_{ij}(x))$ is a positive definite hermitian matrix $forall xin X$) is the kahler form associated to a kahler structure. Supposing $alpha$ a kahler class i want to show that there is a $tinmathbb{R}$ such that $alpha + t beta$ is not a kahler class. Since $betaneq0$ i can find a $tinmathbb{R}$ such that $alpha + t beta$ is not positive definite any more, now i want to prove that there is no form $omega in H^{(1,1)}(X)cap H^2(X,mathbb{R})$ such that $omega=dlambda$ with $lambda$ a real 1-form and $omega=overline{partial}mu$ with $mu$ a complex (1,0)-form (what i'd like to prove is: correcting representatives of cohomology classes with an exact form i don't get a kahler class). From $partialoverline{partial}$-lemma and a little work i know that $omega=ipartialoverline{partial}f$ with f a real function. And now (and here i can't go on) i want to prove that i can't have a function f such that $alpha + t beta+ipartialoverline{partial}f$ is positive definite.

Please, if i made mistakes, or you know how to go on, or another way to solve this, tell me.

Thank you in advance.